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Mirrors > Home > ILE Home > Th. List > exmidontri2or | Unicode version |
Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.) |
Ref | Expression |
---|---|
exmidontri2or | EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidontriim 7160 | . . 3 EXMID | |
2 | onelss 4347 | . . . . . . . 8 | |
3 | 2 | adantl 275 | . . . . . . 7 |
4 | orc 702 | . . . . . . 7 | |
5 | 3, 4 | syl6 33 | . . . . . 6 |
6 | eqimss 3182 | . . . . . . . 8 | |
7 | 6, 4 | syl 14 | . . . . . . 7 |
8 | 7 | a1i 9 | . . . . . 6 |
9 | onelss 4347 | . . . . . . . 8 | |
10 | 9 | adantr 274 | . . . . . . 7 |
11 | olc 701 | . . . . . . 7 | |
12 | 10, 11 | syl6 33 | . . . . . 6 |
13 | 5, 8, 12 | 3jaod 1286 | . . . . 5 |
14 | 13 | ralimdva 2524 | . . . 4 |
15 | 14 | ralimia 2518 | . . 3 |
16 | 1, 15 | syl 14 | . 2 EXMID |
17 | ontri2orexmidim 4531 | . . . 4 DECID | |
18 | 17 | adantr 274 | . . 3 DECID |
19 | 18 | exmid1dc 4161 | . 2 EXMID |
20 | 16, 19 | impbii 125 | 1 EXMID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3o 962 wceq 1335 wcel 2128 wral 2435 wss 3102 c0 3394 csn 3560 EXMIDwem 4155 con0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3773 df-tr 4063 df-exmid 4156 df-iord 4326 df-on 4328 df-suc 4331 |
This theorem is referenced by: onntri52 7179 |
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